Summary: The Daily Mail misreported the findings. We do not know if there is a causal link.
The study they reported on is this one:
The study did not attempt to show causation, and could not do so, as it was based on a survey, not a randomised controlled experiment. The conclusions they drew were limited to correlations, not causation. This is suggestive of further research directions:
consumption of aspartame is associated with greater obesity-related impairments in glucose tolerance.
In conclusion, aspartame artificial sweetener consumption may be associated with greater glucose intolerance, particularly for those with obesity. Future research is needed to determine if from an obesity and diabetes perspective, it may be prudent to limit all sweetener consumption.
The Daily Mail overstepped when it made claims like:
But, new study has shown they increase a person's risk of type 2 diabetes
It is not difficult to come up with plausible confounding variables, which make it impossible to soundly make the causation claim. For example, they do not appear to have considered whether the subjects have already been diagnosed with Type II diabetes, and have modified their diets to include aspartame to help manage their sugar levels.
Similarly, it seems like that obese people (who are more likely to have greater obesity-related impairments) are more likely to be dieting with aspartame, pushing the causation in the opposite direction. Indeed, given that there is more social pressure on women to reduce weight than men, this passage from the study is suggestive of that:
Individuals consuming artificial sweeteners (aspartame or saccharin) had a subtly higher BMI (28 vs. 27 kg/m2), and were more likely to be female (Table 1, P < 0.05).
Another observation against suggests the causation link may be the other way around:
We observe that aspartame was related to significantly greater impairments in glucose tolerance for individuals with obesity, but not lean individuals. In fact, our results suggest a beneficial effect of aspartame in lean individuals. However, as there are very few lean individuals in the population who reported consuming aspartame, this requires further investigation.